The noncentral chi-squared distribution is a generalization of the Chi Squared
Distribution. If Xi are ν independent, normally distributed random
variables with means μi and variances σi2, then the random variable

is distributed according to the noncentral chi-squared distribution.

The noncentral chi-squared distribution has two parameters: ν which specifies
the number of degrees of freedom (i.e. the number of Xi), and λ which is
related to the mean of the random variables Xi by:

(Note that some references define λ as one half of the above sum).

This leads to a PDF of:

where f(x;k) is the central chi-squared distribution
PDF, and Iv(x) is a modified Bessel function of
the first kind.

The following graph illustrates how the distribution changes for different
values of λ:

The following table shows the peak errors (in units of epsilon)
found on various platforms with various floating-point types, along with
comparisons to the R-2.5.1 Math
library. Unless otherwise specified, any floating-point type
that is narrower than the one shown will have effectively
zero error.

Table 17. Errors In CDF of the Noncentral Chi-Squared

Significand Size

Platform and Compiler

ν,λ < 200

ν,λ > 200

53

Win32, Visual C++ 8

Peak=50 Mean=9.9

R Peak=685 Mean=109

Peak=9780 Mean=718

R Peak=3x108 Mean=2x107

64

RedHat Linux IA32, gcc-4.1.1

Peak=270 Mean=27

Peak=7900 Mean=900

64

Redhat Linux IA64, gcc-3.4.4

Peak=107 Mean=17

Peak=5000 Mean=630

113

HPUX IA64, aCC A.06.06

Peak=270 Mean=20

Peak=4600 Mean=560

Error rates for the complement of the CDF and for the quantile functions
are broadly similar. Special mention should go to the mode
function: there is no closed form for this function, so it is evaluated
numerically by finding the maxima of the PDF: in principal this can not
produce an accuracy greater than the square root of the machine epsilon.

There are two sets of test data used to verify this implementation: firstly
we can compare with published data, for example with Table 6 of "Self-Validating
Computations of Probabilities for Selected Central and Noncentral Univariate
Probability Functions", Morgan C. Wang and William J. Kennedy, Journal
of the American Statistical Association, Vol. 89, No. 427. (Sep., 1994),
pp. 878-887. Secondly, we have tables of test data, computed with this
implementation and using interval arithmetic - this data should be accurate
to at least 50 decimal digits - and is the used for our accuracy tests.

First we determine which of the two values (the CDF or its complement)
is likely to be the smaller: for this we can use the relation due to
Temme (see "Asymptotic and Numerical Aspects of the Noncentral Chi-Square
Distribution", N. M. Temme, Computers Math. Applic. Vol 25, No.
5, 55-63, 1993) that:

F(ν,λ;ν+λ) ≈ 0.5

and so compute the CDF when the random variable is less than ν+λ, and its
complement when the random variable is greater than ν+λ. If necessary the
computed result is then subtracted from 1 to give the desired result
(the CDF or its complement).

For small values of the non centrality parameter, the CDF is computed
using the method of Ding (see "Algorithm AS 275: Computing the Non-Central
#2 Distribution Function", Cherng G. Ding, Applied Statistics, Vol.
41, No. 2. (1992), pp. 478-482). This uses the following series representation:

which requires just one call to gamma_p_derivative
with the subsequent terms being computed by recursion as shown above.

For larger values of the non-centrality parameter, Ding's method can
take an unreasonable number of terms before convergence is achieved.
Furthermore, the largest term is not the first term, so in extreme cases
the first term may be zero, leading to a zero result, even though the
true value may be non-zero.

Therefore, when the non-centrality parameter is greater than 200, the
method due to Krishnamoorthy (see "Computing discrete mixtures of
continuous distributions: noncentral chisquare, noncentral t and the
distribution of the square of the sample multiple correlation coefficient",
Denise Benton and K. Krishnamoorthy, Computational Statistics & Data
Analysis, 43, (2003), 249-267) is used.

This method uses the well known sum:

Where Pa(x) is the incomplete gamma function.

The method starts at the λth term, which is where the Poisson weighting
function achieves its maximum value, although this is not necessarily
the largest overall term. Subsequent terms are calculated via the normal
recurrence relations for the incomplete gamma function, and iteration
proceeds both forwards and backwards until sufficient precision has been
achieved. It should be noted that recurrence in the forwards direction
of Pa(x) is numerically unstable. However, since we always start after
the largest term in the series, numeric instability is introduced more
slowly than the series converges.

Computation of the complement of the CDF uses an extension of Krishnamoorthy's
method, given that:

we can again start at the λ'th term and proceed in both directions from
there until the required precision is achieved. This time it is backwards
recursion on the incomplete gamma function Qa(x) which is unstable. However,
as long as we start well before the largest term,
this is not an issue in practice.

The PDF is computed directly using the relation:

Where f(x; v) is the PDF of the central Chi
Squared Distribution and Iv(x) is a modified
Bessel function, see cyl_bessel_i.
For small values of the non-centrality parameter the relation in terms
of cyl_bessel_i
is used. However, this method fails for large values of the non-centrality
parameter, so in that case the infinite sum is evaluated using the method
of Benton and Krishnamoorthy, and the usual recurrence relations for
successive terms.

The quantile functions are computed by numeric inversion of the CDF.

There is no closed
form for the mode of the noncentral chi-squared distribution:
it is computed numerically by finding the maximum of the PDF. Likewise,
the median is computed numerically via the quantile.

The remaining non-member functions use the following formulas:

Some analytic properties of noncentral distributions (particularly unimodality,
and monotonicity of their modes) are surveyed and summarized by: